On the Fock representation of the central extensions of the Heisenberg algebra

We examine the possibility of a direct Fock representation of the recently obtained non-trivial central extensions CEHeis of the Heisenberg algebra, generated by elements a, a†, h and E satisfying the commutation relations [a, a†]CEHeis = h, [h, a†]CEHeis = z E and [a, h]CEHeis = ¯z E, where a and a† are dual, h is self-adjoint, E is the non-zero selfadjoint central element and z 2 C \ {0}. We define the exponential vectors associated with the CEHeis Fock space, we compute their Leibniz function (inner product), we describe the action of a, a† and h on the exponential vectors and we compute the moment generating and characteristic functions of the classical random variable corresponding to the self-adjoint operator X = a + a† + h.